Representations over diagrams of abelian categories II: Abelian model structures

TL;DR

This paper develops a framework for constructing abelian model structures on categories of representations over diagrams of abelian categories, with applications to Gorenstein homological algebra.

Contribution

It introduces a new method to amalgamate compatible families of abelian model categories into a single structure focusing on morphism classes, and describes cofibrant objects explicitly.

Findings

Constructed Gorenstein injective and flat model structures on presheaves of modules.

Provided explicit descriptions of cofibrant objects in the model categories.

Characterized Gorenstein homological objects within the new framework.

Abstract

This is the second paper in a series on representations over diagrams of abelian categories. We show that, under certain conditions, a compatible family of abelian model categories indexed by a skeletal small category can be amalgamated into an abelian model structure on the category of representations. Our approach focuses on classes of morphisms rather than cotorsion pairs of objects. Additionally, we provide an explicit description of cofibrant objects in the resulting abelian model category. As applications, we construct Gorenstein injective and Gorenstein flat model structures on the category of presheaves of modules over a special class of index category and characterize Gorenstein homological objects within this framework.